ECE 476 Power System Analysis Lecture 22

Transient Stability

Alejandro D. Dominguez-Garcia Department of Electrical and Computer Engineering

University of Illinois at Urbana-Champaign

Transient Stability Analysis

  For transient stability analysis we need to consider three systems

1.  Prefault - before the fault occurs the system is assumed to be at an equilibrium point

2.  Faulted - the fault changes the system equations, moving the system away from its equilibrium point

3.  Postfault - after fault is cleared the system hopefully returns to a new operating point

Transient Stability Solution Methods

  There are two methods for solving the transient stability problem

1.  Numerical integration   this is by far the most common technique, particularly

for large systems; during the fault and after the fault the power system differential equations are solved using numerical methods

2.  Direct or energy methods; for a two bus system this method is known as the equal area criteria   mostly used to provide an intuitive insight into the

transient stability problem

SMIB Example

 Assume a generator is supplying power to an infinite bus through two parallel transmission lines. Then a balanced three phase fault occurs at the terminal of one of the lines. The fault is cleared by the opening of this line’s circuit breakers.

SMIB Example, cont’d

Simplified prefault system

SMIB Example, Faulted System

During the fault the system changes

The equivalent system during the fault is then During this fault no power can be transferred from the generator to the system

SMIB Example, Post Fault System

After the fault the system again changes

The equivalent system after the fault is then

SMIB Example, Dynamics

Transient Stability Solution Methods

  There are two methods for solving the transient stability problem

1.  Numerical integration   this is by far the most common technique, particularly

for large systems; during the fault and after the fault the power system differential equations are solved using numerical methods

2.  Direct or energy methods; for a two bus system this method is known as the equal area criteria   mostly used to provide an intuitive insight into the

transient stability problem

Transient Stability Analysis

•  For transient stability analysis we need to consider three systems

1.  Prefault - before the fault occurs the system is assumed to be at an equilibrium point

2.  Faulted - the fault changes the system equations, moving the system away from its equilibrium point

3.  Postfault - after fault is cleared the system hopefully returns to a new operating point

Transient Stability Solution Methods

•  There are two methods for solving the transient stability problem

1.  Numerical integration   this is by far the most common technique, particularly

for large systems; during the fault and after the fault the power system differential equations are solved using numerical methods

2.  Direct or energy methods; for a two bus system this method is known as the equal area criteria   mostly used to provide an intuitive insight into the

transient stability problem

Numerical Integration of DEs

Examples

Euler’s Method

Euler’s Method Algorithm

Euler’s Method Example 1

Euler’s Method Example 1, cont’d

t xactual(t) x(t) Dt=0.1 x(t) Dt=0.05

0 10 10 10

0.1 9.048 9 9.02

0.2 8.187 8.10 8.15

0.3 7.408 7.29 7.35

… … … …

1.0 3.678 3.49 3.58

… … … …

2.0 1.353 1.22 1.29

Euler’s Method Example 2

Euler's Method Example 2, cont'd

Euler's Method Example 2, cont'd

t x1actual(t) x1(t) Dt=0.25

0 1 1

0.25 0.9689 1

0.50 0.8776 0.9375

0.75 0.7317 0.8125

1.00 0.5403 0.6289

… … …

10.0 -0.8391 -3.129

100.0 0.8623 -151,983

Euler's Method Example 2, cont'd

Dt x1(10)

actual -0.8391

0.25 -3.129

0.10 -1.4088

0.01 -0.8823

0.001 -0.8423

Below is a comparison of the solution values for x1(t) at time t = 10 seconds

Transient Stability Example

•  A 60 Hz generator is supplying 550 MW to an infinite bus (with 1.0 per unit voltage) through two parallel transmission lines. Determine initial angle change for a fault midway down one of the lines.���H = 20 seconds, D = 0.1. Use Dt=0.01 second.

Ea

Transient Stability Example, cont'd

Transient Stability Example, cont'd

Transient Stability Example, cont'd

Transient Stability Example, cont'd

Equal Area Criteria

•  The goal of the equal area criteria is to try to determine whether a system is stable or not without having to completely integrate the system response.

System will be stable after the fault if the Decel Area is greater than the Accel. Area